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42 Answers
42

I would love to see more differential geometry offered in undergrad. The course I envision would start with a review of vector calculus, move to studying hypersurfaces in $\mathbb{R}^n$, and then move into a study of manifolds. You could tie all these subjects together via the Fundamental Theorem of Curves, the Fundamental Theorem of Hypersurfaces, and the Fundamental Theorem of Riemannian Geometry. I feel such a course would help bridge the gap between undergrad and grad school; simultaneously reviewing the key ideas of calculus at a high level while also giving a solid foundation from which to study manifolds in grad school.

P.S. I am not a differential geometer. I study homotopy theory. I just thought a course like this was missing from the curriculum. Any feedback on other topics that could be covered or tangents that could be mentioned leading off from this material would be welcome.

I like the idea of promoting mathematics to students, i.e. more explanation of the contribution of mathematics to civilization, so that students have some language and background to justify their subject. See other articles on my web page.

On specific subjects, I have enjoyed showing first year UK students the power of the symbolic algebra packages on dealing with Grobner bases, i.e. solving polynomial equations in more than one variable. To more advanced students one can give exercises like:

Find a polynomial in x,y which has more than 5 critical points, classify them as max, min, saddle, and use the computer algebra package to draw your function and display or indicate the critical points. Verify with the package that the points you find are critical points.

(The last part gives useful lessons in rounding accuracy.) The whole exercise gives students a nice sense of power, as the machine manipulates vast expressions!

When I was in my first year, I always missed order theory. I teached it to myself then and thought many times "Why didn't they teach us this - we would understand everything so much better!"

And I still think so today. Order theory starts off easy, when you lern about relations, preorders, lattices and so on, and then you get into Zorns Lemma, Schörder-Bernstein Theorem and stuff like that.

But I'm also on the category theory track. I think the notion of category will become, just like the notion of a group, more common sense, not just in mathematics but also in computer science, physics, chemistry and maybe even more.

Computing with Feynman diagrams in zero dimension, i.e., a graphical calculus
for tensor contractions. It is very elementary yet can lead quickly into rather deep
mathematics. It would make later studies in say mathematical physics or low dimensional
topology much more congenial.
Possible applications could be

What should one teach to liberal-arts students who will take only one math course, and that because it's required of them?

The conventional answer: Partial fractions. And various useless clerical skills that they'll need if they take second-year calculus, although they'll never take first-year calculus. Et cetera.

My answer: the truth.

E.g. in third grade you were told that
$$
3+3+3+3+3 = 5+5+5
$$
and so on. Why should that be true? Assign that as a homework problem. At this point they may think that means there's some formula to plug this information into to get the answer. They've been taught that memorizing algorithms and applying them is what math is. That's a lie. We should stop lying and level with them.

I posed the question and asked for a course, not a rant on the substance of mathematics education for non-mathematicians: -1
–
Michael HoffmanJun 15 '10 at 23:48

4

There is no "the truth". It's much easier to blindly criticize than to offer constructive answers: -1.
–
André HenriquesMay 11 '11 at 18:02

1

+1, if only for displaying the eternal problem about why is $3+3+3+3+3=5+5+5$. I've asked that of many bright undergrad before, and rarely got and answer better than "that's the law" (the commutative law, they mean, which we all abide by except when we do powers).
–
Dror Bar-NatanMay 11 '11 at 18:56

2

Michael: answering the questions you want to pose is surely something better done on a blog. Redefining the original question to one you can then be righteous about does not strike me as very courteous
–
Yemon ChoiMay 12 '11 at 23:37

Once students have been exposed to linear algebra and vector calculus, build calculus on manifolds using many examples; i.e. go from real $\mathbb{R}$-abstract multilinear to the de Rham complex, illustrating in $\mathbb{R}^3$. All that easen differential geometry, differential topology Riemannian geometry, etc.

But, please, with motivation. Understanding manifolds is THE hardest part of "elementary" advanced mathematics. Just introducing them by an abstract definition and then doing proofs by "local-global" handwaving doesn't do the job; the students will neither have an idea what the definition signifies, nor why the handwaving is allowed. Maybe some non-Euclidean geometry as a nontrivial motivating example would be useful...
–
darij grinbergMar 15 '10 at 11:18

2

I actually would love to see non-euclidean geometry (from a more elementary perspective) and differential geometry unified into one course. You always see elementary books which give you bits and pieces of the idea that there are other geometries, and that the sophisticated way of doing this is differential geometry - then much later in life you get thrown into a course where you pick up from multivariable calculus and define manifolds, tangent spaces, tensor fields, etc... There should be an attempt to show the connection between the two.
–
David CorwinJun 16 '10 at 8:12

What about "Mathematics with Computers"? Having modern computer algebra and symbolic computation tools available, one can use them to present and explore nontrivial examples in various fields of mathematics. Part of the course could also present basic algorithms and other techniques used.

Personally, I think the answer to this question is largely going to depend on one's particularly interests (whether they lie in algebra, analysis, topology, or whatever). This can be seen from many of the previous posts.

That being said, I do think that more number theory would be a great addition to the undergraduate curriculum. Many students take an introductory number theory course (or skip it because they learned it all in high school) and then don't do any more. There are lots of great areas of number theory which don't require too much background. P-adics would be great (Gouvea even laments in his book that p-adics aren't taught earlier - so maybe such a course should use his book). One could teach a basic semester of algebraic number theory, or a course in elliptic curves (following Silverman and Tate, for example). Both of these require no more than a basic course in undergraduate algebra. You can probably find these courses at many top universities, but they usually aren't emphasized as much to undergraduates. The reason why I think that these would be good is because number theory is a particularly beautiful area of math, and by getting glimpses of modern number theory early on, students get to see how beautiful is the math that's ahead of them.
(Another possibility is to have a course on Ireland and Rosen's book A Classical Introduction to Modern Number Theory. Princeton had a junior seminar on this book, for example.)

I also think Riemann surfaces are a very beautiful topic which should be taught early on and aren't too complicated in their most basic form. For, you get to see the deep geometrical theory lying behind the $e^{2 i \pi}=1$ and the ambiguity of complex square roots which you learned about when you were younger. It shows the student that there can be very deep ideas lying behind a simple observation, and it shows the beauty and deep understanding that modern mathematics can lead you to.

First, statistics, indeed, is not taught enough. I studied statistics in a good school, but when it came to actually using it, found that I don't understand it. Second, motivation: they have to show the student how much and how urgently (s)he will need these concepts while on the workplace, with good real examples.